Monopole Floer Homology, Eigenform Multiplicities, and the Seifert–Weber Dodecahedral Space

Abstract

Abstract We show that the Seifert–Weber dodecahedral space $\textsf{SW}$ is a monopole Floer homology $L$-space. The proof relies on our approach to study Floer homology using hyperbolic geometry. Although $\textsf{SW}$ is significantly larger than previous manifolds studied with this technique, we overcome computational complexity issues inherent to our method by exploiting the many symmetries of $\textsf{SW}$. In particular, we prove that small eigenvalues on coexact $1$-forms on $\textsf{SW}$ have large multiplicity.